Photonic crystal

The opal in this bracelet contains a natural periodic microstructure responsible for its iridescent color. It is essentially a natural photonic crystal.

A photonic crystal is a periodic optical nanostructure that affects the motion of photons in much the same way that ionic lattices affect electrons in solids. Photonic crystals occur in nature in the form of structural coloration—and, in different forms, promise to be useful in a range of applications.

In 1887 the English physicist Lord Rayleigh experimented with periodic multi-layer dielectric stacks, showing they had a photonic band-gap in one dimension. Research interest grew with work in 1987 by Yablonovitch and John on periodic optical structures with more than one dimension—now called photonic crystals.

Photonic crystals can be fabricated for one, two, or three dimensions. One-dimensional photonic crystals can be made of layers deposited or stuck together. Two-dimensional ones can be made by photolithography, or by drilling holes in a suitable substrate. Fabrication methods for three-dimensional ones include drilling under different angles, stacking multiple 2-D layers on top of each other, direct laser writing, or, for example, instigating self-assembly of spheres in a matrix and dissolving the spheres.

Photonic crystals can, in principle, find uses wherever light must be manipulated. Existing applications include thin-film optics with coatings for lenses. Two-dimensional photonic-crystal fibers are used in nonlinear devices and to guide exotic wavelengths. Three-dimensional crystals may one day be used in optical computers.

Introduction

Photonic crystals are composed of periodic dielectric, metallo-dielectric—or even superconductor microstructures or nanostructures that affect electromagnetic wave propagation in the same way that the periodic potential in a semiconductor crystal affects electron motion by defining allowed and forbidden electronic energy bands. Photonic crystals contain regularly repeating regions of high and low dielectric constant. Photons (behaving as waves) either propagate through this structure or not, depending on their wavelength. Wavelengths that propagate are called modes, and groups of allowed modes form bands. Disallowed bands of wavelengths are called photonic band gaps. This gives rise to distinct optical phenomena, such as inhibition of spontaneous emission, high-reflecting omni-directional mirrors, and low-loss-waveguiding.

The periodicity of the photonic crystal structure must be around half the wavelength of the electromagnetic waves to be diffracted. This is ~200 nm (blue) to 350 nm (red) for photonic crystals that operate in the visible part of the spectrum—or even less, depending on average index of refraction. The repeating regions of high and low dielectric constant must, therefore, be fabricated at this scale, which is difficult.

History of photonic crystals

Photonic crystals have been studied in one form or another since 1887, but no one used the term photonic crystal until over 100 years later—after Eli Yablonovitch and Sajeev John published two milestone papers on photonic crystals in 1987.[1][2]

Before 1987, one-dimensional photonic crystals in the form of periodic multi-layer dielectric stacks (such as the Bragg mirror) were studied extensively. Lord Rayleigh started their study in 1887,[3] by showing that such systems have a one-dimensional photonic band-gap, a spectral range of large reflectivity, known as a stop-band. Today, such structures are used in a diverse range of applications—from reflective coatings to enhancing LED efficiency to highly reflective mirrors in certain laser cavities (see, for example, VCSEL). A detailed theoretical study of one-dimensional optical structures was performed by Vladimir P. Bykov,[4] who was the first to investigate the effect of a photonic band-gap on the spontaneous emission from atoms and molecules embedded within the photonic structure. Bykov also speculated as to what could happen if two- or three-dimensional periodic optical structures were used.[5] The concept of three-dimensional photonic crystals was then discussed by Ohtaka in 1979,[6] who also developed a formalism for the calculation of the photonic band structure. However, these ideas did not take off until after the publication of two milestone papers in 1987 by Yablonovitch and John. Both these papers concerned high-dimensional periodic optical structures, i.e., photonic crystals. Yablonovitch's main goal was to engineer photonic density of states to control the spontaneous emission of materials embedded in the photonic crystal. John's idea was to use photonic crystals to affect localisation and control of light.

After 1987, the number of research papers concerning photonic crystals began to grow exponentially. However, due to the difficulty of fabricating these structures at optical scales (see Fabrication challenges), early studies were either theoretical or in the microwave regime, where photonic crystals can be built on the more accessible centimetre scale. (This fact is due to a property of the electromagnetic fields known as scale invariance. In essence, electromagnetic fields, as the solutions to Maxwell's equations, have no natural length scale—so solutions for centimetre scale structure at microwave frequencies are the same as for nanometre scale structures at optical frequencies.)

By 1991, Yablonovitch had demonstrated the first three-dimensional photonic band-gap in the microwave regime.[7] The structure that Yablonvitch was able to produce involved drilling an array of holes in a transparent material, where the holes of each layer form an inverse diamond structure – today it is known as Yablonovite.

In 1996, Thomas Krauss demonstrated a two-dimensional photonic crystal at optical wavelengths.[8] This opened the way to fabricate photonic crystals in semiconductor materials by borrowing methods from the semiconductor industry.

Today, such techniques use photonic crystal slabs, which are two dimensional photonic crystals "etched" into slabs of semiconductor. Total internal reflection confines light to the slab, and allows photonic crystal effects, such as engineering photonic dispersion in the slab. Researchers around the world are looking for ways to use photonic crystal slabs in integrated computer chips, to improve optical processing of communications—both on-chip and between chips.

Such techniques have yet to mature into commercial applications, but two-dimensional photonic crystals are commercially used in photonic crystal fibres[9] (otherwise known as holey fibres, because of the air holes that run through them). Photonic crystal fibres were first developed by Philip Russell in 1998, and can be designed to possess enhanced properties over (normal) optical fibres.

Study has proceeded more slowly in three-dimensional than in two-dimensional photonic crystals. This is because of more difficult fabrication.[9] Three-dimensional photonic crystal fabrication had no inheritable semiconductor industry techniques to draw on. Attempts have been made, however, to adapt some of the same techniques, and quite advanced examples have been demonstrated,[10] for example in the construction of "woodpile" structures constructed on a planar layer-by-layer basis. Another strand of research has tried to construct three-dimensional photonic structures from self-assembly—essentially letting a mixture of dielectric nano-spheres settle from solution into three-dimensionally periodic structures that have photonic band-gaps. Vasily Astratov's group from the Ioffe Institute realized in 1995 that natural and synthetic opals are photonic crystals with an incomplete bandgap.[11] The first demonstration of an "inverse opal" structure with a complete photonic bandgap came in 2000, from researchers at the University of Toronto, Canada.[12] The ever expanding field of biomimetics—the study of natural structures to better understand and use them in design—is also helping researchers in photonic crystals.[13][14] For example, in 2006 a naturally-occurring photonic crystal was discovered in the scales of a Brazilian beetle.[15]

Construction strategies

The fabrication method depends on the number of dimensions that the photonic bandgap must exist in.

One-dimensional photonic crystals

In a one-dimensional photonic crystal, layers of different dielectric constant may be deposited or adhered together to form a band gap in a single direction. A Bragg grating is an example of this type of photonic crystal. One-dimensional photonic crystals can be either isotropic or anisotropic, with the latter having potential use as an optical switch.[16]

One-dimensional photonic crystal can form as an infinite number of parallel alternating layers filled with a metamaterial and vacuum.[17] This produced identical PBG structures for TE and TM modes.

Recently, researchers fabricated a graphene-based Bragg grating (one-dimensional photonic crystal) and demonstrated that it supports excitation of surface electromagnetic waves in the periodic structure by using 633 nm He-Ne laser as the light source.[18] Besides, a novel type of one-dimensional graphene-dielectric photonic crystal has also been proposed. This structure can act as a far-IR filter and can support low-loss surface plasmons for waveguide and sensing applications.[19]

Two-dimensional photonic crystals

In two dimensions, holes may be drilled in a substrate that is transparent to the wavelength of radiation that the bandgap is designed to block. Triangular and square lattices of holes have been successfully employed.

The Holey fiber or photonic crystal fiber can be made by taking cylindrical rods of glass in hexagonal lattice, and then heating and stretching them, the triangle-like airgaps between the glass rods become the holes that confine the modes.

Three-dimensional photonic crystals

There are several structure types that have been constructed:[20]

Fabrication challenges

Higher-dimensional photonic crystal fabrication faces two major challenges:

One promising fabrication method for two-dimensionally periodic photonic crystals is a photonic-crystal fiber, such as a holey fiber. Using fiber draw techniques developed for communications fiber it meets these two requirements, and photonic crystal fibres are commercially available. Another promising method for developing two-dimensional photonic crystals is the so-called photonic crystal slab. These structures consist of a slab of material—such as silicon—that can be patterned using techniques from the semiconductor industry. Such chips offer the potential to combine photonic processing with electronic processing on a single chip.

For three dimensional photonic crystals, various techniques have been used—including photolithography and etching techniques similar to those used for integrated circuits.[10] Some of these techniques are already commercially available. To avoid the complex machinery of nanotechnological methods, some alternate approaches involve growing photonic crystals from colloidal crystals as self-assembled structures.

Mass-scale 3D photonic crystal films and fibres can now be produced using a shear-assembly technique that stacks 200–300 nm colloidal polymer spheres into perfect films of fcc lattice. Because the particles have a softer transparent rubber coating, the films can be stretched and molded, tuning the photonic bandgaps and producing striking structural color effects.

Computing photonic band structure

The photonic band gap (PBG) is, essentially, the gap between the air-line and the dielectric-line in the dispersion relation of the PBG system. To design photonic crystal systems, it is essential to engineer the location and size of the bandgap by computational modeling using any of the following methods:

Essentially, these methods solve for the frequencies (normal models) of the photonic crystal for each value of the propagation direction given by the wave vector, or vice versa. The various lines in the band structure, correspond to the different cases of n, the band index. For an introduction to photonic band structure, see Joannopoulos.[26]

Band structure of a 1D Photonic Crystal, DBR air-core calculated using plane wave expansion technique with 101 planewaves, for d/a=0.8, and dielectric contrast of 12.250.

The plane wave expansion method can be used to calculate the band structure using an eigen formulation of the Maxwell's equations, and thus solving for the eigen frequencies for each of the propagation directions, of the wave vectors. It directly solves for the dispersion diagram. Electric field strength values can also be calculated over the spatial domain of the problem using the eigen vectors of the same problem. For the picture shown to the right, corresponds to the band-structure of a 1D distributed Bragg reflector (DBR) with air-core interleaved with a dielectric material of relative permittivity 12.25, and a lattice period to air-core thickness ratio (d/a) of 0.8, is solved using 101 planewaves over the first irreducible Brillouin zone.

To speed calculation of the frequency band structure, the Reduced Bloch Mode Expansion (RBME) method can be used.[27] The RBME method applies "on top" of any of the primary expansion methods mentioned above. For large unit cell models, the RBME method can reduce time for computing the band structure by up to two orders of magnitude.

Applications

Photonic crystals are attractive optical materials for controlling and manipulating light flow. One dimensional photonic crystals are already in widespread use, in the form of thin-film optics, with applications from low and high reflection coatings on lenses and mirrors to colour changing paints and inks. Higher-dimensional photonic crystals are of great interest for both fundamental and applied research, and the two dimensional ones are beginning to find commercial applications.

The first commercial products involving two-dimensionally periodic photonic crystals are already available in the form of photonic-crystal fibers, which use a microscale structure to confine light with radically different characteristics compared to conventional optical fiber for applications in nonlinear devices and guiding exotic wavelengths. The three-dimensional counterparts are still far from commercialization but may offer additional features such as optical nonlinearity required for the operation of optical transistors used in optical computers, when some technological aspects such as manufacturability and principal difficulties such as disorder are under control.

See also

References

  1. ↑ E. Yablonovitch (1987), "Inhibited Spontaneous Emission in Solid-State Physics and Electronics" (PDF), Physical Review Letters, 58 (20): 2059–2062, Bibcode:1987PhRvL..58.2059Y, doi:10.1103/PhysRevLett.58.2059, PMID 10034639
  2. ↑ S. John (1987), "Strong localization of photons in certain disordered dielectric superlattices" (PDF), Physical Review Letters, 58 (23): 2486–2489, Bibcode:1987PhRvL..58.2486J, doi:10.1103/PhysRevLett.58.2486, PMID 10034761
  3. ↑ J. W. S. Rayleigh (1888), "On the remarkable phenomenon of crystalline reflexion described by Prof. Stokes" (PDF), Phil. Mag, 26: 256–265, doi:10.1080/14786448808628259
  4. ↑ V. P. Bykov (1972), "Spontaneous Emission in a Periodic Structure", Soviet Journal of Experimental and Theoretical Physics, 35: 269–273, Bibcode:1972JETP...35..269B
  5. ↑ V. P. Bykov (1975), "Spontaneous emission from a medium with a band spectrum", Quantum Electronics, 4 (7): 861–871, Bibcode:1975QuEle...4..861B, doi:10.1070/QE1975v004n07ABEH009654
  6. ↑ K. Ohtaka (1979), "Energy band of photons and low-energy photon diffraction", Physical Review B, 19 (10): 5057–5067, Bibcode:1979PhRvB..19.5057O, doi:10.1103/PhysRevB.19.5057
  7. ↑ E. Yablonovitch, T.J. Gmitter, K.M. Leung, E; Gmitter, TJ; Leung, KM (1991), "Photonic band structure: the face-centered-cubic case employing nonspherical atoms" (PDF), Physical Review Letters, 67 (17): 2295–2298, Bibcode:1991PhRvL..67.2295Y, doi:10.1103/PhysRevLett.67.2295, PMID 10044390
  8. ↑ T. F. Krauss, R. M. DeLaRue, S. Brand; Rue; Brand (1996), "Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths", Nature, 383 (6602): 699–702, Bibcode:1996Natur.383..699K, doi:10.1038/383699a0
  9. 1 2 Jennifer Ouellette (2002), "Seeing the Future in Photonic Crystals" (PDF), The Industrial Physicist, 7 (6): 14–17, archived from the original (PDF) on August 12, 2011
  10. 1 2 Review: S. Johnson (MIT) Lecture 3: Fabrication technologies for 3d photonic crystals, a survey
  11. ↑ Astratov, VN; Bogomolov, VN; Kaplyanskii, AA; Prokofiev, AV; Samoilovich, LA; Samoilovich, SM; Vlasov, Yu A (1995), "Optical spectroscopy of opal matrices with CdS embedded in its pores: Quantum confinement and photonic band gap effects", Il Nuovo Cimento D, 17 (11-12): 1349–1354, Bibcode:1995NCimD..17.1349A, doi:10.1007/bf02457208
  12. ↑ Blanco, Alvaro; Blanco, Alvaro; Chomski, Emmanuel; Grabtchak, Serguei; Ibisate, Marta; Leonard, Stephen W.; Lopez, Cefe; Meseguer, Francisco; et al. (2000), "Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres", Nature, 405 (6785): 437–440, doi:10.1038/35013024, PMID 10839534
  13. ↑ Kolle, Mathias (2011), Photonic Structures Inspired by Nature (1st ed.), ISBN 978-3-642-15168-2
  14. ↑ Ross C. McPhedran & Andrew R. Parker (2015), Biomimetics: Lessons on optics from nature’s school
  15. ↑ Michael H. Bartl; Richey, Lauren R.; Gardner, John S.; Cha, Jennifer N.; Bartl, Michael H.; et al. (2008), "Discovery of a diamond-based photonic crystal structure in beetle scales", Physical Review Letters, 77 (5): 50904, Bibcode:2008PhRvE..77e0904G, doi:10.1103/PhysRevE.77.050904
  16. ↑ Fiddy, Michael A.; Schenk, John O.; Cao, Yang (2008), "Third Order Nonlinear Effect near a Degenerate Band Edge", Optics and Photonics Letters, 1 (1): 1–7, doi:10.1142/S1793528808000033
  17. ↑ Pravdin, K. V.; Popov, I. Yu. (2014). "Photonic crystal with negative index material layers" (PDF). Nanosystems: physics, chemistry, mathematics. 5 (5): 626–643.
  18. ↑ K.V.Sreekanth; Zeng, Shuwen; Shang, Jingzhi; Yong, Ken-Tye; Yu, Ting; et al. (2012), "Excitation of surface electromagnetic waves in a graphene-based Bragg grating", Scientific Reports, 2: 737, Bibcode:2012NatSR...2E.737S, doi:10.1038/srep00737, PMC 3471096Freely accessible, PMID 23071901
  19. ↑ . H. Hajian; Soltani-Vala, A.; Kalafi, M.; et al. (2013), "Characteristics of band structure and surface plasmons supported by a one-dimensional graphene-dielectric photonic crystal", Optics Communications, 292: 149, Bibcode:2013OptCo.292..149H, doi:10.1016/j.optcom.2012.12.002
  20. ↑ http://ab-initio.mit.edu/book/photonic-crystals-book.pdf
  21. ↑
  22. ↑ Turner, M. D.; Saba, M.; Zhang, Q.; Cumming, B. P.; Schröder-Turk, G. E.; Gu, M. (2013). "Miniature chiral beamsplitter based on gyroid photonic crystals". Nature Photonics. 7 (10): 801. Bibcode:2013NaPho...7..801T. doi:10.1038/nphoton.2013.233.
  23. ↑ P. Ordejon (1998), "Order-n tight-binding methods for electronic-structure and molecular dynamics", Computational materials science, 12 (3): 157–191, doi:10.1016/S0927-0256(98)00027-5
  24. ↑ Richard M Martin, Linear Scaling 'Order-N' Methods in Electronic Structure Theory
  25. ↑ http://emlab.utep.edu/ee5390em21.htm
  26. ↑ John D Joannopoulos, Johnson SG, Winn JN & Meade RD (2008), Photonic Crystals: Molding the Flow of Light (2nd ed.), Princeton NJ: Princeton University Press, ISBN 978-0-691-12456-8
  27. ↑ M.I. Hussein (2009), "Reduced Bloch mode expansion for periodic media band structure calculations", Proceedings of the Royal Society A, 465 (2109): 2825–2848, arXiv:0807.2612Freely accessible, Bibcode:2009RSPSA.465.2825H, doi:10.1098/rspa.2008.0471

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